In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
10. y = ln(t^(3/2))

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Recognize that the function is given as \(y = \ln\left(t^{\frac{3}{2}}\right)\), which is a natural logarithm of a power function.

Use the logarithmic property that allows you to bring the exponent in front: \(\ln\left(t^{\frac{3}{2}}\right) = \frac{3}{2} \ln(t)\).

Rewrite the function as \(y = \frac{3}{2} \ln(t)\) to simplify differentiation.

Recall the derivative of \(\ln(t)\) with respect to \(t\) is \(\frac{1}{t}\), so apply the constant multiple rule to get \(\frac{dy}{dt} = \frac{3}{2} \cdot \frac{1}{t}\).

Express the derivative as \(\frac{dy}{dt} = \frac{3}{2t}\), which is the derivative of the original function with respect to \(t\).

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Logarithmic Differentiation

Logarithmic differentiation involves applying the derivative rules to functions involving logarithms. The derivative of ln(u), where u is a function of the variable, is (1/u) times the derivative of u. This technique simplifies differentiation of functions expressed as logarithms.

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Logarithmic properties, like ln(a^b) = b * ln(a), allow simplification of expressions before differentiation. Applying these properties can transform complex logarithmic functions into simpler forms, making differentiation more straightforward.

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